A number of interesting examples can now be produced via manipulation of Cohen forcings. If \(X\)  is a Polish space then the Cohen forcing \(P_X\)  is the poset of nonempty open subsets of \(X\)  ordered by inclusion. It adds a point \(\dot{x}\in X\) , the unique point in the intersection of open sets in the generic filter. It is not difficult to see [1]} that if \(f\colon X\rightarrow Y\)  is a continuous open map between Polish spaces then \(P_X\Vdash \dot{f}(\dot{x})\)  is a point \(P_Y\) -generic over the ground model.
